Um... starting sentences with words other than "I" would be a good start. Not to sound like a smart-ass, but I don't quite see what the problem is. I'm writing a novel in first person at the moment, so I see where you're coming from, but it's a really simple solution.

Focus less on what the character is doing. What I'm picturing is your story looking like this:

I walk into the room. I see my friends sitting at a table. I say hello to them. I walk over and take a seat. I take a drink of water from the pitcher.

If this is an accurate representation, then you can easily edit your writing to be less focused on the narrator, and more on the world around them.

I walk into the room. My friends are sitting at a table.
"Hello," I say, and I take a seat. John pours me a glass of water and I drink it.

It's not that difficult to switch it around. But, more importantly, if all you're doing is focusing on the character's actions, the "and then I did this, and then I did this, and then I did this," it's going to get boring and rhythmic and repetitive really quickly.

To avoid this, I think you should focus on the character. Start the sentence with the word which is most important for the character. Again, you can start sentence with the word which he is doing. Thus you can avoid "I".

I know what you mean. When I started writing first-person narratives, I took a step back and looked at how choppy it felt saying "I this" and "I that."

Generally, I found that when I write my first draft in first-person I start dozens of sentences with "I felt like" or "I realized that". When I go back to edit, I eliminate most of these and instead, like Infinity Man said, focus on other people or the scene at hand, etc, etc. Or just rewrite it. But don't let this stop you from writing in the first place. Just be mindful of it and know that when you finish the first draft, you can always go back and fix it.

I find that when I write in first person, I start a lot of sentences with 'I'. How can I avoid this?

Same is the case here, but I don't think that's a problem since I had submitted a chapter of my novel in a writing community and wasn't really pointed out for using many I-s
But if you're not satisfied with it, you can try writing in third person but I won't say it's a good option, you can consider it if you like but I felt being kicked out of my own novel when I tried writing in third person.